Coinduction All the Way Up

Coinduction All the Way Up
Damien Pous
Plume group, CNRS, ENS Lyon
Gallium, Paris, 12.o9.2o16
Program Equivalence
Through equational reasoning
!a. a = !(a | a)
= !a | !a
= !a
(a. a = a | a)
(!(P | Q) = !P | !Q)
(!P | !P = !P)
(Axiomatic, syntax-based, inductive)
Damien Pous, CNRS, ENS Lyon
2/11
Program Equivalence
Through bisimulations
R , {hP | Q, Q | Pi | ∀P, Q}
(Behavioural, LTS-based, coinductive)
Damien Pous, CNRS, ENS Lyon
3/11
Program Equivalence
Through bisimulations
R , {hP | Q, Q | Pi | ∀P, Q}
P|Q
τ
(νa)(P 0 | Q 0 )
Damien Pous, CNRS, ENS Lyon
R
R
/
Q|P
τ
(νa)(Q 0 | P 0 )
3/11
Program Equivalence
Through bisimulations
R , {h(ν ã)(P | Q), (ν ã)(Q | P) | ∀ã, P, Qi}
P|Q
τ
(νa)(P 0 | Q 0 )
Damien Pous, CNRS, ENS Lyon
R
R
Q|P
τ
(νa)(Q 0 | P 0 )
3/11
Program Equivalence
Through bisimulations up-to
R , {hP | Q, Q | Pi | ∀P, Q}
P|Q
τ
(νa)(P 0 | Q 0 )
R
R
/
Q|P
τ
(νa)(Q 0 | P 0 )
(νa)(R)
Damien Pous, CNRS, ENS Lyon
3/11
Program Equivalence
Through bisimulations up-to
R , {h!(P | Q), !P | !Qi}
Damien Pous, CNRS, ENS Lyon
4/11
Program Equivalence
Through bisimulations up-to
R , {h!(P | Q), !P | !Qi}
!(P | Q)
α
!(P | Q) |
Damien Pous, CNRS, ENS Lyon
(P 0
R
| Q) R
/
!P | !Q
α
(!P | P 0 ) | !Q
4/11
Program Equivalence
Through bisimulations up-to
R , {h!(P | Q), !P | !Qi}
!(P | Q)
α
!(P | Q) |
(P 0
R
| Q) R
/
!P | !Q
α
(!P | P 0 ) | !Q
∼
(!P | P 0 ) | (!Q | Q)
Damien Pous, CNRS, ENS Lyon
4/11
Program Equivalence
Through bisimulations up-to
R , {h!(P | Q), !P | !Qi}
!(P | Q)
α
!(P | Q) |
(P 0
R
| Q) R
/
!P | !Q
α
(!P | P 0 ) | !Q
∼
(!P | P 0 ) | (!Q | Q)
∼
(!P | !Q) | (P 0 | Q)
Damien Pous, CNRS, ENS Lyon
4/11
Program Equivalence
Through bisimulations up-to
R , {h!(P | Q), !P | !Qi}
!(P | Q)
α
!(P | Q) |
(P 0
R
/
| Q) R
!P | !Q
α
(!P | P 0 ) | !Q
∼
C(R)
Damien Pous, CNRS, ENS Lyon
(!P | P 0 ) | (!Q | Q)
∼
(!P | !Q) | (P 0 | Q)
4/11
Program Equivalence
Through bisimulations up-to
R , {h!(P | Q), !P | !Qi}
!(P | Q)
α
!(P | Q) |
(P 0
R
/
| Q) R
!P | !Q
α
(!P | P 0 ) | !Q
∼
C(R)
(!P | P 0 ) | (!Q | Q)
∼
(!P | !Q) | (P 0 | Q)
A mix of
•
behavioural, LTS-based, coinductive
•
axiomatic, syntax-based, inductive
Damien Pous, CNRS, ENS Lyon
4/11
Abstract coinduction
Let b a monotone function on a complete lattice
•
bisimulation
candidate
Coinduction
x ≤ y ≤ b(y )
law to be
proved
Damien Pous, CNRS, ENS Lyon
x ≤ νb
program
equivalence
5/11
Abstract coinduction
Let b a monotone function on a complete lattice
•
bisimulation
candidate
Coinduction
x ≤ y ≤ b(y )
law to be
proved
•
x ≤ νb
program
equivalence
an up-to
technique
Coinduction up to
x ≤ y ≤ b( f (y ))
x ≤ νb
Damien Pous, CNRS, ENS Lyon
5/11
Up-to techniques
bisimulation
candidate
up-to
technique
x ≤ y ≤ b( f (y ))
law to be
proved
x ≤ νb
program
equivalence
The function f can be
•
sound: it just makes the rule valid
•
respectful: f ◦ (b ∩ id) ≤ b ◦ f
•
compatible: f ◦ b ≤ b ◦ f
[Sangiorgi’94]
[me’07]
The latter two classes are closed under union and composition
Damien Pous, CNRS, ENS Lyon
6/11
The companion
Let t be the largest compatible function
•
We have
1.
2.
3.
4.
5.
id ≤ t and t ◦ t ≤ t, i.e., t is a closure
νb = t(⊥)
νb = t(νb)
νb = ν(b ◦ t)
t coincides with the largest respectful
•
Intuitively t(x) is “what can be deduced assuming x”
•
Leads to a new presentation of parameterized coinduction
Damien Pous, CNRS, ENS Lyon
7/11
Parameterized Coinduction (Up-to)
y ≤ t(⊥)
y ≤ νb
Init
y≤ x
y ≤ t(x)
f ≤t
y ≤ t(x)
program
equivalence
law to be
proved
y ≤ f (t(x))
Up to f
reason inductively, using
some valid up-to technique
y ≤ b(t( y ∪ x))
Done
use coinduction
hypotheses
Damien Pous, CNRS, ENS Lyon
y ≤ t(x)
CoInd
assume y by
coinduction
8/11
Second order techniques
The companion is itself a coinductive object
t = νB
(for some B : (X → X ) → (X → X ) whose post-fixpoints are the
compatible functions)
Damien Pous, CNRS, ENS Lyon
9/11
Second order techniques
The companion is itself a coinductive object
t = νB
(for some B : (X → X ) → (X → X ) whose post-fixpoints are the
compatible functions)
We can apply the previous results
potential
generalisation
companion of B
f ≤ g ≤ B( T (g ))
up-to technique
to be proved valid
Damien Pous, CNRS, ENS Lyon
f ≤ t
companion of b
9/11
Back to process algebra
[Standard approach] The following function is respectful:
C : R 7→ {hC [P̃], C [Q̃]i | P̃ R Q̃}
Damien Pous, CNRS, ENS Lyon
10/11
Back to process algebra
[Standard approach] The following function is respectful:
C : R 7→ {hC [P̃], C [Q̃]i | P̃ R Q̃}
[New approach] The following functions are below t:
c(νa) : R 7→ {h(νa)P, (νa)Qi | P R Q}
c| : R 7→ {hP1 | P2 , Q1 | Q2 i | Pi R Qi }
c! : R 7→ {h!P, !Qi | P R Q}
...
(A routine check in each case, thanks to the second order
companion)
Damien Pous, CNRS, ENS Lyon
10/11
Back to process algebra
[Standard approach] The following function is respectful:
C : R 7→ {hC [P̃], C [Q̃]i | P̃ R Q̃}
[New approach] The following functions are below t:
c(νa) : R 7→ {h(νa)P, (νa)Qi | P R Q}
c| : R 7→ {hP1 | P2 , Q1 | Q2 i | Pi R Qi }
c! : R 7→ {h!P, !Qi | P R Q}
...
(A routine check in each case, thanks to the second order
companion)
Damien Pous, CNRS, ENS Lyon
10/11
Summary
Huge simplification of the theory of up-to techniques. . .
. . . just by focusing on the largest one
•
parameterized coinduction through the companion
•
second order techniques for the companion
•
straightforward formalisation in Coq
In progress: categorical account
Damien Pous, CNRS, ENS Lyon
11/11